We recapitulate here the Hartree-Fock (HF) theory at finite temperature
[1]}.
The partition function and the statistical density matrix
at the temperature \(\beta ^{-1}=k_B T\)  are
in the form, \(Z=\mathrm {tr}\left[ e^{-\beta \hat{H}^{\prime }} \right]\) ,
and \(\hat{D}=e^{-\beta \hat{H}^{\prime }}/Z\) , respectively,
where \(\hat{H}^{\prime }\equiv \hat{H}-\mu \hat{N}\)  with the one-body HF Hamiltonian
\(\hat{H}\) , the particle number operator \(\hat{N}\) , and the chemical potential \(\mu \) .
The one-body density matrix is given as
\(\rho _{ij}=\mathrm {tr}\left[ \hat{D} \hat{c}_j^\dagger \hat{c}_i \right]= \sum _\alpha \langle i | \alpha  \rangle f_{\beta \mu }(\epsilon _\alpha )\langle \alpha  | j \rangle \) 
